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210 IX ANALYTIC FUNCTIONS
PROBLEMS |
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1. (a) Let P(#i, . . . , ur+i) be a polynomial with coefficients in K, «i, . . . , ocp elements of
K such that |aj| < 1 for 1 ^ / ^ r. Suppose there exists a ball B in K of center 0 and a scalar function / analytic in B and such that f(z) = P(z,/(aiz), . . . ,/(arz)) for every z e B. Show that / can be extended to a function g analytic in the whole set K and satisfying the same functional equation in K (use (9.4.2)).
(b) Suppose K = C, and suppose there is a real number ft and a scalar func-
tion / analytic for &(z) > ft, and satisfying in that subset the equation f(z) = P(r,/(z-fai), ...,/(z + aP)), where the at are complex numbers with ^(at)>Q. Show that /can be extended to a function g analytic in C and satisfying the same functional equation.
(c) Generalize the preceding results to functions of any number of variables.
2. Let D be a connected open set in Cp, D' the image of D by the mapping
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Let / be a complex function analytic in D, and suppose D O Rp is not empty,
D n D' is connected and / takes real values in D n Rp. Show that / can be ex- tended to a function g analytic in D u D'. (Consider in D' the function (zlf . . . , Zp) ->/(zlf . . . , Zp), and use (9.4.4).) |
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5. EXAMPLES OF ANALYTIC FUNCTIONS; THE EXPONENTIAL
FUNCTION; THE NUMBER n
(9.5.1) Let P(z), Q(z) be two polynomials in Kp, such that Q is not identically
0; then P(z)/Q(z) is analytic in the (open) set of the points z such that Q(z) ^ 0 (i.e., the set of points where the function is defined).
It is obvious that any polynomial is an entire function. By (9.3.2) all
we have to do is to show that 1/z is analytic for z ^ 0; but if z0 ^ 0, we can write
1= 1
z
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-- +1
ZQ ZQ ZQ ZQ
where the power series is absolutely summable for |z — z0| < |z0|. Q.E.D.
Consider now the function e* of the real variable x; we prove it is an
entire function. From the Taylor formula (8.14.3) we derive, for any n (using Section 8.8)
, „ x x2 xn
e, = 1 + _ + _ + ...+_
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